A quadrilateral with 4 right angles is a rectangle. A rectangle is automatically a parallelogram, hence opposite sides are equal and parallel.
A square is a quadrilateral with right angles and 4 sides the same length. A square is automatically a rectangle. The unit square is a square whose sides have length 1. If your plane is described by coordinates, the unit square usually runs from 0 to 1, in the x and y directions.
A rhombus is a quadrilateral whose sides are all the same length. A rhombus is automatically a parallelogram. Some people call a rhombus a diamond.
Sometimes it helps to remember that a parallelogram is a pushed-over rectangle, and a rhombus is a pushed-over square.
A kite is a convex quadrilateral with two adjacent sides the same length and the other two adjacent sides the same length. Stand it up on the sharpest point and it looks like a kite. Cut the kite down the middle with a vertical line and the resulting triangles are congruent by SSS. Thus the left and right angles of a kite are congruent.
In a trapezoid, the top and bottom are parallel. In an isosceles trapezoid, the left and right sides are equal in length, and the top is, by convention, shorter than the bottom. (Some people use the term trapezoid to mean an isosceles trapezoid.) Drop perpendiculars from the left and right corners down to the base. Since the top and bottom are parallel, these vertical segments have the same length. We already know the left and right sides have the same length, so the left and right triangles are congruent by LH. This means the base angles are congruent. Use the properties of alternate interior angles to show the two angles at the top of the isosceles trapezoid are congruent.